Conditions for Existence of Fourier Series (Dirichlet Conditions)

Summary

In this lecture, the concept of Fourier series expansion is explored, explaining its use for analyzing periodic signals. The video discusses the conditions for the existence of Fourier series, known as Dirichlet conditions, which include having a finite number of maxima and minima, a finite number of discontinuities, and being absolutely integrable over one period. Examples are provided to illustrate these conditions, clarifying why some periodic signals may not have a Fourier series. The lecture also differentiates between periodic and non-periodic signals, with the latter requiring Fourier transform analysis, a topic for a subsequent chapter.

Takeaways

• 📚 Fourier series is used for analyzing periodic signals, while Fourier transform is used for non-periodic signals.
• 🔍 The existence of Fourier series is subject to certain conditions, known as Dirichlet conditions, named after the German mathematician Gerhard Dirichlet.
• 📈 Condition 1: A periodic signal must have a finite number of maxima and minima over one period for a Fourier series to exist.
• 📉 Condition 2: A periodic signal must have a finite number of discontinuities over one period for a Fourier series to exist.
• 📊 Condition 3: A signal must be absolutely integrable over one period, meaning the integral of the signal over one period must be finite.
• 🚫 Non-periodic signals inherently do not have Fourier series because they do not repeat in a regular pattern.
• ❌ A signal with infinite maxima, minima, or discontinuities over one period violates the conditions for a Fourier series.
• ✅ Examples were provided to illustrate how to determine if a signal meets the conditions for a Fourier series, such as a signal with one maximum and minimum per period.
• ❓ The script clarifies potential confusion regarding the term 'absolutely integrable' in the context of periodic signals, which are never energy signals but can be integrable over a finite period.
• 💡 The lecture emphasizes the importance of understanding these conditions to correctly apply Fourier series in signal analysis.

Q & A

What is the primary purpose of Fourier series expansion?

The primary purpose of Fourier series expansion is to analyze periodic signals. It is a tool used to decompose a periodic function into a sum of sine and cosine functions.

What is the difference between Fourier series and Fourier transform?

Fourier series is used for analyzing periodic signals, while Fourier transform is used for analyzing non-periodic signals. The conditions for their existence are different and are discussed separately in the lectures.

Who established the conditions for the existence of Fourier series?

The conditions for the existence of Fourier series were given by the German mathematician Johann Carl Friedrich Gauss, also known as Gerhard Chalet.

What is the first condition for the existence of Fourier series?

The first condition states that the number of maxima and minima over the range of the time period of a periodic signal should be finite.

Can you give an example of a signal where the Fourier series expansion does not exist due to the first condition?

An example of a signal where the Fourier series expansion does not exist due to the first condition is a signal with infinite maxima and minima within one period, such as a signal with a structure that repeats but has an increasing number of peaks and troughs.

What is the second condition for the existence of Fourier series?

The second condition states that a periodic signal should have a finite number of discontinuities over the range of the time period.

How does the presence of infinite discontinuities affect the existence of a Fourier series?

If a signal has infinite discontinuities within one period, the Fourier series expansion for that signal will not exist because it violates the second condition for the existence of Fourier series.

What does it mean for a signal to be absolutely integrable over the range of a time period?

A signal is absolutely integrable over the range of a time period if the integral of the signal's absolute value over one period results in a finite value.

Why is the concept of absolute integrability important for Fourier series?

The concept of absolute integrability is important for Fourier series because it ensures that the integral of the signal over one period is finite, which is a requirement for the existence of a Fourier series according to the third condition.

What happens if a signal is not absolutely integrable over the range of a time period?

If a signal is not absolutely integrable over the range of a time period, the integral of the signal over that period will be infinite, and thus the Fourier series expansion for the signal will not exist.

Can a periodic signal be an energy signal?

No, a periodic signal cannot be an energy signal because the integral of a periodic signal from minus infinity to infinity would result in an infinite area, which is a characteristic of power signals, not energy signals.